《结构动力学》课程教学课件（讲稿）10 Evaluation of structural-property matrices

Wuhan University of TechnologyChapter10Evaluation of structural-propertymatrices10.1 Elastic properties10.2 Mass properties10.3 Damping properties10.4 External loading10.5Geometricstiffness10.6 Choice of property formulation10-1

10-1 Wuhan University of Technology 10.1 Elastic properties 10.2 Mass properties 10.3 Damping properties 10.4 External loading 10.5 Geometric stiffness 10.6 Choice of property formulation Chapter 10 Evaluation of structural-property matrices

Wuhan Universityof Technology10.1 Elastic propertiesP.=I1P = 1FIGURE10-1Definitionofflexibilityinfluencecoefficients.Thedefinition ofaflexibilityinfluence coefficient isdeflectionofcoordinateiduetounitloadappliedtocoordinatej10-2

10-2 Wuhan University of Technology 10.1 Elastic properties FIGURE 10-1 Definition of flexibility influence coefficients. The definition of a flexibility influence coefficient is

Wuhan Universityof Technology10.1 Elastic propertiesThe deflection at point 1 due to any combination of loads may be expressed asU1=fiiP+fi2Pe+fisPa+..+finPnf12fi3..[Ji1fiNP11f21fo2fosfaNV2P2预fi,f.f.NViPv=fpv=ffs10-3

10-3 Wuhan University of Technology 10.1 Elastic properties The deflection at point 1 due to any combination of loads may be expressed as

Wuhan Universityof Technology10.1 Elastic propertiesThestiffness influencecoefficients inFig.102arenumericallyequal totheapplied forces required to maintain the specified displacement condition. Theyarepositivewhenthesenseof theappliedforcecorrespondstoapositivedisplacementandnegativeotherwise.P,=kup,=kPinD.-2=k2P,=k,Pn=kN2FFIGURE10-2Definitionofstiffnessinfluencecoefficients.10-4

10-4 Wuhan University of Technology 10.1 Elastic properties The stiffness influence coefficients in Fig. 102 are numerically equal to the applied forces required to maintain the specified displacement condition. They are positive when the sense of the applied force corresponds to a positive displacement and negative otherwise. FIGURE 10-2 Definition of stiffness influence coefficients

Wuhan Universityof Technology10.1 Elastic propertiesStrainenergy.Thestrainenergystoredinanystructuremaybeexpressedconvenientlyintermsof eithertheflexibilityorthestiffnessmatrix.ThestrainenergyUisequaltotheworkdoneindistortingthesystem;thusN1ZUP,Ui2i=11Ufpp121vTkvU12Finally,whenitisnotedthatthestrainenergystoredinastablestructureduring any distortion must always be positive, it is evident thatvTkv>0pTfp>0and10-5

10-5 Wuhan University of Technology 10.1 Elastic properties Strain energy . The strain energy stored in any structure may be expressed conveniently in terms of either the flexibility or the stiffness matrix. The strain energy U is equal to the work done in distorting the system; thus Finally, when it is noted that the strain energy stored in a stable structure during any distortion must always be positive, it is evident that

Wuhan University of Technology10.1 Elastic propertiesMatriceswhich satisfythis condition,where v or pis any arbitrary nonzero vectoraresaidtobepositivedefinite;positivedefinitematrices(andconsequentlytheflexibilityandstiffnessmatricesofa stable structure)arenonsingularand canbeinverted.InvertingthestiffnessmatrixandpremultiplyingbothsidesofEq.(96)bytheinverseleadstoKwhichuponcomparisonwithEg.(105)demonstratesthattheflexibilitymatrixistheinverseofthestiffnessmatrixk-1 =f10-6

10-6 Wuhan University of Technology 10.1 Elastic properties Matrices which satisfy this condition, where v or p is any arbitrary nonzero vector, are said to be positive definite; positive definite matrices (and consequently the flexibility and stiffness matrices of a stable structure) are nonsingular and can be inverted. Inverting the stiffness matrix and premultiplying both sides of Eq. (96) by the inverse leads to which upon comparison with Eq. (105) demonstrates that the flexibility matrix is the inverse of the stiffness matrix:

WuhanUniversityof Technology1o.1 ElasticpropertiesBetti'slaw.Apropertywhichisveryimportant instructuraldynamicsanalysiscanbederivedbyapplyingtwosetsof loadstoastructureinreversesequenceandcomparingexpressionsfortheworkdoneinthetwocases.Consider,forexample,thetwodifferentloadsystemsandtheirresultingdisplacementsshownin Fig. 103. If the loads a are applied first followed by loads b, the work done willbeasfollows:Load system a:Load system b:PlaP2aP3aPib-P2b-P3bDeflections b:Deflections a:U3bUlaU2aU3aUih-U2b10-7

10-7 Wuhan University of Technology 10.1 Elastic properties Betti's law. A property which is very important in structuraldynamics analysis can be derived by applying two sets of loads to a structure in reverse sequence and comparing expressions for the work done in the two cases. Consider, for example, the two different load systems and their resulting displacements shown in Fig. 103. If the loads a are applied first followed by loads b, the work done will be as follows:

Wuhan University of TechnologyC10.1 Elastic propertiesCase 1:Waa=EpiaUia=IpTvaWhb + Wab = p,Tv+p.TvbWi = Waa +Wbb+Wab=p.Tva+p,Tv+ pTvCase 2:Wab=p,TvWaa+Wba=p.Tva+pTvaW2=Whb+ Waa+Wba=pTv +pTva+p,Tv10-8

10-8 Wuhan University of Technology 10.1 Elastic properties Case 1: Case 2:

WuhanUniversityof Technology10.1 ElasticpropertiesThedeformationofthe structureisindependentof theloadingsequence,however;thereforethestrainenergyandhencealsotheworkdonebytheloadsis the same in both these cases; that is, Wi = W2. From a comparison of Eqs.(1011)and (1012) it maybe concluded that Wab= Wbai thusTVp=PbEquation(1013)isanexpressionofBetti'slaw;itstatesthattheworkdonebyonesetof loadsonthedeflectionsduetoasecondsetofloadsisequaltotheworkofthesecondsetofloadsactingonthedeflectionsduetothefirst.=pTfpp.Tfp.10-9

10-9 Wuhan University of Technology 10.1 Elastic properties The deformation of the structure is independent of the loading sequence, however; therefore the strain energy and hence also the work done by the loads is the same in both these cases; that is, W1 = W 2. From a comparison of Eqs. (1011) and (1012) it may be concluded that Wab = Wba; thus Equation (1013) is an expression of Betti's law; it states that the work done by one set of loads on the deflections due to a second set of loads is equal to the work of the second set of loads acting on the deflections due to the first

Wuhan University of Technology10.1 Elastic propertiesItis evidentthatf=fThus the flexibility matrix must be symmetric; that is, f, = fi,.. This is anexpressionofMaxwell'slawofreciprocaldeflections.SubstitutingsimilarlywithEq.(96)(andnotingthatp=f.)leadstok = kT10-10

10-10 Wuhan University of Technology 10.1 Elastic properties It is evident that Thus the flexibility matrix must be symmetric; that is, . This is an expression of Maxwell's law of reciprocal deflections. Substituting similarly with Eq. (96) (and noting that p = fS) leads to